A Short Account of the History of Mathematics · W. W. Rouse Ball
In the last chapter but one I discussed the development of European mathematics to a date which corresponds roughly with the end of the "dark ages"; and in the last chapter I traced the history of the mathematics of the Indians and Arabs to the same date. The mathematics of the two or three centuries that follow and are treated in this chapter are characterised by the introduction of the Arab mathematical text-books and of Greek books derived from Arab sources, and the assimilation of the new ideas thus presented.
It was, however, from Spain, and not from Arabia, that a knowledge of eastern mathematics first came into western Europe. The Moors had established their rule in Spain in 747, and by the tenth or eleventh century had attained a high degree of civilisation. Though their political relations with the caliphs at Bagdad were somewhat unfriendly, they gave a ready welcome to the works of the great Arab mathematicians. In this way the Arab translations of the writings of Euclid, Archimedes, Apollonius, Ptolemy, and perhaps of other Greek authors, together with the works of the Arabian algebraists, were read and commented on at the three great Moorish schools of Granada, Cordova, and Seville. It seems probable that these works indicate the full extent of Moorish learning, but, as all knowledge was jealously guarded from Christians, it is impossible to speak with certainty either on this point or on that of the time when the Arab books were first introduced into Spain.
The eleventh century. The earliest Moorish writer of distinction of whom I find mention is Geber ibn Aphla, who was born at Seville and died towards the latter part of the eleventh century at Cordova. He wrote on astronomy and trigonometry, and was acquainted with the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides.
Arzachel. Another Arab of about the same date was Arzachel, who was living at Toledo in 1080. He suggested that the planets moved in ellipses, but his contemporaries with scientific intolerance declined to argue about a statement which was contrary to Ptolemy's conclusions in the Almagest.
The twelfth century. During the course of the twelfth century copies of the books used in Spain were obtained in western Christendom. The first step towards procuring a knowledge of Arab and Moorish science was taken by an English monk, Adelhard of Bath,AB On the influence of Adelhard and Ben Ezra, see the "Abhandlungen zur Geschichte der Mathematik" in the Zeitschrift für Mathematik, vol. xxv, 1880. who, under the disguise of a Mohammedan student, attended some lectures at Cordova about 1120 and obtained a copy of Euclid's Elements. This copy, translated into Latin, was the foundation of all the editions known in Europe till 1533, when the Greek text was recovered. How rapidly a knowledge of the work spread we may judge when we recollect that before the end of the thirteenth century Roger Bacon was familiar with it, while before the close of the fourteenth century the first five books formed part of the regular curriculum at many universities. The enunciations of Euclid seem to have been known before Adelhard's time, and possibly as early as the year 1000, though copies were rare. Adelhard also issued a text-book on the use of the abacus.
Ben Ezra.AB During the same century other translations of the Arab text-books or commentaries on them were obtained. Amongst those who were most influential in introducing Moorish learning into Europe I may mention Abraham Ben Ezra. Ben Ezra was born at Toledo in 1097, and died at Rome in 1167. He was one of the most distinguished Jewish rabbis who had settled in Spain, where it must be recollected that they were tolerated and even protected by the Moors on account of their medical skill. Besides some astronomical tables and an astrology, Ben Ezra wrote an arithmetic; in this he explains the Arab system of numeration with nine symbols and a zero, gives the fundamental processes of arithmetic, and explains the rule of three.
Gerard. Another European who was induced by the reputation of the Arab schools to go to Toledo was Gerard, who was born at Cremona in 1114 and died in 1187. He translated the Arab edition of the Almagest, the works of Alhazen, and the works of Alfarabius, whose name is otherwise unknown to us: it is believed that the Arabic numerals were used in this translation, made in 1136, of Ptolemy's work. Gerard also wrote a short treatise on algorism which exists in manuscript in the Bodleian Library at Oxford. He was acquainted with one of the Arab editions of Euclid's Elements, which he translated into Latin.
John Hispalensis. Among the contemporaries of Gerard was John Hispalensis of Seville, originally a rabbi, but converted to Christianity and baptized under the name given above. He made translations of several Arab and Moorish works, and also wrote an algorism which contains the earliest examples of the extraction of the square roots of numbers by the aid of the decimal notation.
The thirteenth century. During the thirteenth century there was a revival of learning throughout Europe, but the new learning was, I believe, confined to a very limited class. The early years of this century are memorable for the development of several universities, and for the appearance of three remarkable mathematicians--Leonardo of Pisa, Jordanus, and Roger Bacon, the Franciscan monk of Oxford. Henceforward it is to Europeans that we have to look for the development of mathematics, but until the invention of printing the knowledge was confined to a very limited class.
Leonardo. Leonardo Fibonacci (i.e. filius Bonaccii) generally known as Leonardo of Pisa, was born at Pisa about 1175. His father Bonacci was a merchant, and was sent by his fellow-townsmen to control the custom-house at Bugia in Barbary; there Leonardo was educated, and he thus became acquainted with the Arabic or decimal system of numeration, as also with Alkarismi's work on Algebra, which was described in the last chapter. It would seem that Leonardo was entrusted with some duties, in connection with the custom-house, which required him to travel. He returned to Italy about 1200, and in 1202 published a work called Algebra et almuchabala (the title being taken from Alkarismi's work), but generally known as the Liber Abaci. He there explains the Arabic system of numeration, and remarks on its great advantages over the Roman system. He then gives an account of algebra, and points out the convenience of using geometry to get rigid demonstrations of algebraical formulae. He shews how to solve simple equations, solves a few quadratic equations, and states some methods for the solution of indeterminate equations; these rules are illustrated by problems on numbers. The algebra is rhetorical, but in one case letters are employed as algebraical symbols. This work had a wide circulation, and for at least two centuries remained a standard authority from which numerous writers drew their inspiration.
The Liber Abaci is especially interesting in the history of arithmetic, since practically it introduced the use of the Arabic numerals into Christian Europe. The language of Leonardo implies that they were previously unknown to his countrymen; he says that having had to spend some years in Barbary he there learnt the Arabic system, which he found much more convenient than that used in Europe; he therefore published it "in order that the Latin race might no longer be deficient in that knowledge." Now Leonardo had read very widely, and had travelled in Greece, Sicily, and Italy; there is therefore every presumption that the system was not then commonly employed in Europe.
Though Leonardo introduced the use of Arabic numerals into commercial affairs, it is probable that a knowledge of them as current in the East was previously not uncommon among travellers and merchants, for the intercourse between Christians and Mohammedans was sufficiently close for each to learn something of the language and common practices of the other. We can also hardly suppose that the Italian merchants were ignorant of the method of keeping accounts used by some of their best customers; and we must recollect, too, that there were numerous Christians who had escaped or been ransomed after serving the Mohammedans as slaves. It was, however, Leonardo who brought the Arabic system into general use, and by the middle of the thirteenth century a large proportion of the Italian merchants employed it by the side of the old system.
The majority of mathematicians must have already known of the system from the works of Ben Ezra, Gerard, and John Hispalensis. But shortly after the appearance of Leonardo's book Alfonso of Castile (in 1252) published some astronomical tables, founded on observations made in Arabia, which were computed by Arabs, and which, it is generally believed, were expressed in Arabic notation. Alfonso's tables had a wide circulation among men of science, and probably were largely instrumental in bringing these numerals into universal use among mathematicians. By the end of the thirteenth century it was generally assumed that all scientific men would be acquainted with the system: thus Roger Bacon writing in that century recommends algorism (that is, the arithmetic founded on the Arab notation) as a necessary study for theologians who ought, he says, "to abound in the power of numbering." We may then consider that by the year 1300, or at the latest 1350, these numerals were familiar both to mathematicians and to Italian merchants.
So great was Leonardo's reputation that the Emperor Frederick II. stopped at Pisa in 1225 in order to hold a sort of mathematical tournament to test Leonardo's skill, of which he had heard such marvellous accounts. The competitors were informed beforehand of the questions to be asked, some or all of which were composed by John of Palermo, who was one of Frederick's suite. This is the first time that we meet with an instance of those challenges to solve particular problems which were so common in the sixteenth and seventeenth centuries. The first question propounded was to find a number of which the square, when either increased or decreased by 5, would remain a square. Leonardo gave an answer, which is correct, namely 41/12 . The next question was to find by the methods used in the tenth book of Euclid a line whose length x should satisfy the equation x^3 + 2x^2 + 10x = 20 . Leonardo showed by geometry that the problem was impossible, but he gave an approximate value of the root of this equation, namely, 1 . 22' 7" 42"' 33"" 4^v 40^vi , which is equal to 1.3688081075... , and is correct to nine places of decimals. Another question was as follows. Three men, A, B, C, possess a sum of money u , their shares being in the ratio 3:2:1 . A takes away x , keeps half of it, and deposits the remainder with D; B takes away y , keeps two-thirds of it, and deposits the remainder with D; C takes away all that is left, namely z , keeps five-sixths of it, and deposits the remainder with D. This deposit with D is found to belong to A, B, and C in equal proportions. Find u , x , y , and z . Leonardo showed that the problem was indeterminate, and gave as one solution u = 47 , x = 33 , y = 13 , z = 1 . The other competitors failed to solve any of these questions.
The chief work of Leonardo is the Liber Abaci alluded to above. This work contains a proof of the well-known result (a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (bc - ad)^2 = (ad + bc)^2 + (bd - ac)^2. He also wrote a geometry, termed Practica Geometriae, which was issued in 1220. This is a good compilation, and some trigonometry is introduced; among other propositions and examples he finds the area of a triangle in terms of its sides. Subsequently he published a Liber Quadratorum dealing with problems similar to the first of the questions propounded at the tournament. He also issued a tract dealing with determinate algebraical problems: these are all solved by the rule of false assumption in the manner explained above.
Frederick II. The Emperor Frederick II., who was born in 1194, succeeded to the throne in 1210, and died in 1250, was not only interested in science, but did as much as any other single man of the thirteenth century to disseminate a knowledge of the works of the Arab mathematicians in western Europe. The university of Naples remains as a monument of his munificence. I have already mentioned that the presence of the Jews had been tolerated in Spain on account of their medical skill and scientific knowledge, and as a matter of fact the titles of physician and algebraist were for a long time nearly synonymous; thus the Jewish physicians were admirably fitted both to get copies of the Arab works and to translate them. Frederick II. made use of this fact to engage a staff of learned Jews to translate the Arab works which he obtained, though there is no doubt that he gave his patronage to them the more readily because it was singularly offensive to the pope, with whom he was then engaged in a quarrel. At any rate, by the end of the thirteenth century copies of the works of Euclid, Archimedes, Apollonius, Ptolemy, and of several Arab authors were obtainable from this source, and by the end of the next century were not uncommon. From this time, then, we may say that the development of science in Europe was independent of the aid of the Arabian schools.
Jordanus. Among Leonardo's contemporaries was a German mathematician, whose works were until the last few years almost unknown. This was Jordanus Nemorarius, sometimes called Jordanus de Saxonia or Teutonicus. Of the details of his life we know but little, save that he was elected general of the Dominican order in 1222. The works enumerated in the footnote hereto are attributed to him, and if we assume that these works have not been added to or improved by subsequent annotators, we must esteem him one of the most eminent mathematicians of the middle ages.
His knowledge of geometry is illustrated by his De Triangulis and De Isoperimetris. The most important of these is the De Triangulis, which is divided into four books. The first book, besides a few definitions, contains thirteen propositions on triangles which are based on Euclid's Elements. The second book contains nineteen propositions, mainly on the ratios of straight lines and the comparison of the areas of triangles; for example, one problem is to find a point inside a triangle so that the lines joining it to the angular points may divide the triangle into three equal parts. The third book contains twelve propositions mainly concerning arcs and chords of circles. The fourth book contains twenty-eight propositions, partly on regular polygons and partly on miscellaneous questions such as the duplication and trisection problems.
The Algorithmus Demonstratus contains practical rules for the four fundamental processes, and Arabic numerals are generally (but not always) used. It is divided into ten books dealing with properties of numbers, primes, perfect numbers, polygonal numbers, ratios, powers, and the progressions. It would seem from it that Jordanus knew the general expression for the square of any algebraic multinomial.
The De Numeris Datis consists of four books containing solutions of one hundred and fifteen problems. Some of these lead to simple or quadratic equations involving more than one unknown quantity. He shews a knowledge of proportion; but many of the demonstrations of his general propositions are only numerical illustrations of them.
In several of the propositions of the Algorithmus and De Numeris Datis letters are employed to denote both known and unknown quantities, and they are used in the demonstrations of the rules of arithmetic as well as of algebra. As an example of this I quote the following proposition, the object of which is to determine two quantities whose sum and product are known.
Dato numero per duo diuiso si, quod ex ductu unius in alterum producitur, datum fuerit, et utrumque eorum datum esse necesse est.
Sit numerus datus abc diuisus in ab et c, atque ex ab in c fiat d datus, itemque ex abc in se fiat e. Sumatur itaque quadruplum d, qui fit f, quo dempto de e remaneat g, et ipse erit quadratum differentiae ab ad c. Extrahatur ergo radix ex g, et sit h, eritque h differentia ab ad c. cumque sic h datum, erit et c et ab datum.
Huius operatio facile constabit hoc modo. Verbi gratia sit x diuisus in numeros duos, atque ex ductu unius eorum in alium fiat xxi; cuius quadruplum et ipsum est lxxxiiii, tollatur de quadrato x, hoc est c, et remanent xvi, cuius radix extrahatur, quae erit quatuor, et ipse est differentia. Ipsa tollatur de x et reliquum, quod est vi, dimidietur, eritque medietas iii, et ipse est minor portio et maior vii.
It will be noticed that Jordanus, like Diophantus and the Hindoos, denotes addition by juxtaposition. Expressed in modern notation his argument is as follows. Let the numbers be a + b (which I will denote by gamma ) and c . Then gamma + c is given; hence (gamma + c)^2 is known; denote it by e . Again gamma c is given; denote it by d ; hence 4gamma c , which is equal to 4d , is known; denote it by f . Then (gamma - c)^2 is equal to e-f , which is known; denote it by g . Therefore gamma - c = sqrt g , which is known; denote it by h . Hence gamma + c and gamma - c are known, and therefore gamma and c can be at once found. It is curious that he should have taken a sum like a + b for one of his unknowns. In his numerical illustration he takes the sum to be 10 and the product 21.
Save for one instance in Leonardo's writings, the above works are the earliest instances known in European mathematics of syncopated algebra in which letters are used for algebraical symbols. It is probable that the Algorithmus was not generally known until it was printed in 1534, and it is doubtful how far the works of Jordanus exercised any considerable influence on the development of algebra. In fact it constantly happens in the history of mathematics that improvements in notation or method are made long before they are generally adopted or their advantages realized. Thus the same thing may be discovered over and over again, and it is not until the general standard of knowledge requires some such improvement, or it is enforced by some one whose zeal or attainments compel attention, that it is adopted and becomes part of the science. Jordanus in using letters or symbols to represent any quantities which occur in analysis was far in advance of his contemporaries. A similar notation was tentatively introduced by other and later mathematicians, but it was not until it had been thus independently discovered several times that it came into general use.
It is not necessary to describe in detail the mechanics, optics, or astronomy of Jordanus. The treatment of mechanics throughout the middle ages was generally unintelligent.
No mathematicians of the same ability as Leonardo and Jordanus appear in the history of the subject for over two hundred years. Their individual achievements must not be taken to imply the standard of knowledge then current, but their works were accessible to students in the following two centuries, though there were not many who seem to have derived much benefit therefrom, or who attempted to extend the bounds of arithmetic and algebra as there expounded.
During the thirteenth century the most famous centres of learning in western Europe were Paris and Oxford, and I must now refer to the more eminent members of those schools.
Holywood. I will begin by mentioning John de Holywood, whose name is often written in the latinized form of Sacrobosco. Holywood was born in Yorkshire and educated at Oxford; but after taking his master's degree he moved to Paris, and taught there till his death in 1244 or 1246. His lectures on algorism and algebra are the earliest of which I can find mention. His work on arithmetic was for many years a standard authority; it contains rules, but no proofs; it was printed at Paris in 1496. He also wrote a treatise on the sphere, which was made public in 1256: this had a wide and long-continued circulation, and indicates how rapidly a knowledge of mathematics was spreading. Besides these, two pamphlets by him, entitled respectively De Computo Ecclesiastico and De Astrolabio, are still extant.
Roger Bacon. Another contemporary of Leonardo and Jordanus was Roger Bacon, who for physical science did work somewhat analogous to what they did for arithmetic and algebra. Roger Bacon was born near Ilchester in 1214, and died at Oxford on June 11, 1294. He was the son of royalists, most of whose property had been confiscated at the end of the civil wars: at an early age he was entered as a student at Oxford, and is said to have taken orders in 1233. In 1234 he removed to Paris, then the intellectual capital of western Europe, where he lived for some years devoting himself especially to languages and physics; and there he spent on books and experiments all that remained of his family property and his savings. He returned to Oxford soon after 1240, and there for the following ten or twelve years he laboured incessantly, being chiefly occupied in teaching science. His lecture room was crowded, but everything that he earned was spent in buying manuscripts and instruments. He tells us that altogether at Paris and Oxford he spent over £2000 in this way--a sum which represents at least £20,000 nowadays.
Bacon strove hard to replace logic in the university curriculum by mathematical and linguistic studies, but the influences of the age were too strong for him. His glowing eulogy on "divine mathematics" which should form the foundation of a liberal education, and which "alone can purge the intellect and fit the student for the acquirement of all knowledge," fell on deaf ears. We can judge how small was the amount of geometry which was implied in the quadrivium, when he tells us that in geometry few students at Oxford read beyond Euc. i, 5; though we might perhaps have inferred as much from the character of the work of Boethius.
At last worn out, neglected, and ruined, Bacon was persuaded by his friend Grosseteste, the great Bishop of Lincoln, to renounce the world and take the Franciscan vows. The society to which he now found himself confined was singularly uncongenial to him, and he beguiled the time by writing on scientific questions and perhaps lecturing. The superior of the order heard of this, and in 1257 forbade him to lecture or publish anything under penalty of the most severe punishments, and at the same time directed him to take up his residence at Paris, where he could be more closely watched.
Clement IV., when in England, had heard of Bacon's abilities, and in 1266 when he became Pope he invited Bacon to write. The Franciscan order reluctantly permitted him to do so, but they refused him any assistance. With difficulty Bacon obtained sufficient money to get paper and the loan of books, and in the short space of fifteen months he produced in 1267 his Opus Majus with two supplements which summarized what was then known in physical science, and laid down the principles on which it, as well as philosophy and literature, should be studied. He stated as the fundamental principle that the study of natural science must rest solely on experiment; and in the fourth part he explained in detail how astronomy and physical sciences rest ultimately on mathematics, and progress only when their fundamental principles are expressed in a mathematical form. Mathematics, he says, should be regarded as the alphabet of all philosophy.
The results that he arrived at in this and his other works are nearly in accordance with modern ideas, but were too far in advance of that age to be capable of appreciation or perhaps even of comprehension, and it was left for later generations to rediscover his works, and give him that credit which he never experienced in his lifetime. In astronomy he laid down the principles for a reform of the calendar, explained the phenomena of shooting stars, and stated that the Ptolemaic system was unscientific in so far as it rested on the assumption that circular motion was the natural motion of a planet, while the complexity of the explanations required made it improbable that the theory was true. In optics he enunciated the laws of reflexion and in a general way of refraction of light, and used them to give a rough explanation of the rainbow and of magnifying glasses. Most of his experiments in chemistry were directed to the transmutation of metals, and led to no useful results. He gave the composition of gunpowder, but there is no doubt that it was not his own invention, though it is the earliest European mention of it. On the other hand, some of his statements appear to be guesses which are more or less ingenious, while some of them are certainly erroneous.
In the years immediately following the publication of his Opus Majus he wrote numerous works which developed in detail the principles there laid down. Most of these have now been published, but I do not know of the existence of any complete edition. They deal only with applied mathematics and physics.
Clement took no notice of the great work for which he had asked, except to obtain leave for Bacon to return to England. On the death of Clement, the general of the Franciscan order was elected Pope and took the title of Nicholas IV. Bacon's investigations had never been approved of by his superiors, and he was now ordered to return to Paris, where we are told he was immediately accused of magic; he was condemned in 1280 to imprisonment for life, but was released about a year before his death.
Campanus. The only other mathematician of this century whom I need mention is Giovanni Campano, or in the latinized form Campanus, a canon of Paris. A copy of Adelhard's translation of Euclid's Elements fell into the hands of Campanus, who added a commentary thereon in which he discussed the properties of a regular re-entrant pentagon. He also, besides some minor works, wrote the Theory of the Planets, which was a free translation of the Almagest.
The fourteenth century. The history of the fourteenth century, like that of the one preceding it, is mostly concerned with the assimilation of Arab mathematical text-books and of Greek books derived from Arab sources.
Bradwardine. A mathematician of this time, who was perhaps sufficiently influential to justify a mention here, is Thomas Bradwardine, Archbishop of Canterbury. Bradwardine was born at Chichester about 1290. He was educated at Merton College, Oxford, and subsequently lectured in that university. From 1335 to the time of his death he was chiefly occupied with the politics of the church and state; he took a prominent part in the invasion of France, the capture of Calais, and the victory of Cressy. He died at Lambeth in 1349. His mathematical works, which were probably written when he was at Oxford, are the Tractatus de Proportionibus, printed at Paris in 1495; the Arithmetica Speculativa, printed at Paris in 1502; the Geometria Speculativa, printed at Paris in 1511; and the De Quadratura Circuli, printed at Paris in 1495. They probably give a fair idea of the nature of the mathematics then read at an English university.
Oresmus. Nicholas Oresmus was another writer of the fourteenth century. He was born at Caen in 1323, became the confidential adviser of Charles V., by whom he was made tutor to Charles VI., and subsequently was appointed bishop of Lisieux, at which city he died on July 11, 1382. He wrote the Algorismus Proportionum, in which the idea of fractional indices is introduced. He also issued a treatise dealing with questions of coinage and commercial exchange; from the mathematical point of view it is noticeable for the use of vulgar fractions and the introduction of symbols for them.
By the middle of this century Euclidean geometry (as expounded by Campanus) and algorism were fairly familiar to all professed mathematicians, and the Ptolemaic astronomy was also generally known. About this time the almanacks began to add to the explanation of the Arabic symbols the rules of addition, subtraction, multiplication, and division, "de algorismo." The more important calendars and other treatises also inserted a statement of the rules of proportion, illustrated by various practical questions.
In the latter half of this century there was a general revolt of the universities against the intellectual tyranny of the schoolmen. This was largely due to Petrarch, who in his own generation was celebrated as a humanist rather than as a poet, and who exerted all his power to destroy scholasticism and encourage scholarship. The result of these influences on the study of mathematics may be seen in the changes then introduced in the study of the quadrivium. The stimulus came from the university of Paris, where a statute to that effect was passed in 1366, and a year or two later similar regulations were made at other universities; unfortunately no text-books are mentioned. We can, however, form a reasonable estimate of the range of mathematical reading required, by looking at the statutes of the universities of Prague, of Vienna, and of Leipzig.
By the statutes of Prague, dated 1384, candidates for the bachelor's degree were required to have read Holywood's treatise on the sphere, and candidates for the master's degree to be acquainted with the first six books of Euclid, optics, hydrostatics, the theory of the lever, and astronomy. Lectures were actually delivered on arithmetic, the art of reckoning with the fingers, and the algorism of integers; on almanacks, which probably meant elementary astrology; and on the Almagest, that is, on Ptolemaic astronomy. There is, however, some reason for thinking that mathematics received far more attention here than was then usual at other universities.
At Vienna, in 1389, a candidate for a master's degree was required to have read five books of Euclid, common perspective, proportional parts, the measurement of superficies, and the Theory of the Planets. The book last named is the treatise by Campanus which was founded on that by Ptolemy. This was a fairly respectable mathematical standard, but I would remind the reader that there was no such thing as "plucking" in a medieval university. The student had to keep an act or give a lecture on certain subjects, but whether he did it well or badly he got his degree, and it is probable that it was only the few students whose interests were mathematical who really mastered the subjects mentioned above.
The fifteenth century. A few facts gleaned from the history of the fifteenth century tend to shew that the regulations about the study of the quadrivium were not seriously enforced. The lecture lists for the years 1437 and 1438 of the university of Leipzig (founded in 1409, the statutes of which are almost identical with those of Prague as quoted above) are extant, and shew that the only lectures given there on mathematics in those years were confined to astrology. The records of Bologna, Padua, and Pisa seem to imply that there also astrology was the only scientific subject taught in the fifteenth century, and even as late as 1598 the professor of mathematics at Pisa was required to lecture on the Quadripartitum, an astrological work purporting (probably falsely) to have been written by Ptolemy. The only mathematical subjects mentioned in the registers of the university of Oxford as read there between the years 1449 and 1463 were Ptolemy's astronomy, or some commentary on it, and the first two books of Euclid. Whether most students got as far as this is doubtful. It would seem, from an edition of Euclid's Elements published at Paris in 1536, that after 1452 candidates for the master's degree at that university had to take an oath that they had attended lectures on the first six books of that work.
Beldomandi. The only writer of this time that I need mention here is Prodocimo Beldomandi of Padua, born about 1380, who wrote an algoristic arithmetic, published in 1410, which contains the summation of a geometrical series; and some geometrical works.
By the middle of the fifteenth century printing had been introduced, and the facilities it gave for disseminating knowledge were so great as to revolutionize the progress of science. We have now arrived at a time when the results of Arab and Greek science were known in Europe; and this perhaps, then, is as good a date as can be fixed for the close of this period, and the commencement of that of the renaissance. The mathematical history of the renaissance begins with the career of Regiomontanus; but before proceeding with the general history it will be convenient to collect together the chief facts connected with the development of arithmetic during the middle ages and the renaissance. To this the next chapter is devoted.
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